Chapter 18, Problem 44E

(a)

To determine

To find the mean and the standard deviation of the scores.

Answer to Problem 44E

The mean is $\$137.50$ and the standard deviation is $\$148.56$ .

Explanation of Solution

It is given in the question, the table for a museum that offers several levels of membership with the amount of donation. Thus, the mean and the standard deviation of the scores is calculated from the values given in the table as:

  $\begin{array}{l}\mu =E(X)\\ =50(0.41)+100(0.37)+250(0.14)+500(0.07)+1000(0.01)\\ =\$137.50\end{array}$

  $\begin{array}{l}\sigma =\sqrt{{(50-137.5)}^2(0.41)+{(100-137.5)}^2(0.37)+{(250-137.5)}^2(0.14)+{(500-137.5)}^2(0.07)+{(1000-137.5)}^2(0.01)}\\ =\$148.56\end{array}$

(b)

To determine

To explain would you expect the donations for a day to follow a Normal model if during their annual membership drive they hope to sign up $50$ new members each day.

Answer to Problem 44E

No, we do not expect.

Explanation of Solution

It is given in the question, the table for a museum that offers several levels of membership with the amount of donation. And the mean is $\$137.50$ and the standard deviation is $\$148.56$ . Thus, during their annual membership drive they hope to sign up $50$ new members each day so the new members would probably make donations typical of the current member populations, so the distribution would resemble the population, skewed to the right. Therefore, we would not expect the donations for a day to follow a Normal model.

(c)

To determine

To describe the sampling model for these means.

Explanation of Solution

It is given in the question, the table for a museum that offers several levels of membership with the amount of donation. And the mean is $\$137.50$ and the standard deviation is $\$148.56$ . Thus, let us first check the appropriate conditions and assumptions for the following data as:

Random condition: The members are not randomly selected. They are simply the new members that day. However, the donations they make are probably typical of the donations made by current members.

Independent condition: It is reasonable to think that the donations for the new members would not affect one another.

$10\%$ condition: The sample size of $50$ donations represents less than $10\%$ of all scores.

Large enough sample condition: A sample of $50$ donations is large enough thus, it is satisfied.

Now, let us find the mean and the standard deviation as:

  $\begin{array}{l}{\mu }_{\overline{x}}=\$137.50\\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{148.56}{\sqrt{50}}\\ =\$21.01\end{array}$

Now, since the conditions are satisfied the sampling distribution model for the mean of $50$ donations is Normal, with ${\mu }_{\overline{x}}=\$137.50$ and standard deviation ${\sigma }_{\overline{x}}=\$21.01$ .